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Conversion between Degrees and Radians

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Angles in Radians and Degrees
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Most people are familiar with measuring angles in degrees. It is easy to picture angles like 30^\circ 45^\circ or  90^\circ and the fact that  360^\circ makes up an entire circle. Over 2000 years ago the Babylonians used a base 60 number system and divided up a circle into 360 equal parts. This became the standard and it is how most people think of angles today. 

However, there are many units with which to measure angles. For example, the gradian was invented along with the metric system and it divides a circle into 400 equal parts. The sizes of these different units are very arbitrary. 

A radian is a unit of measuring angles that is based on the properties of circles. This makes it more meaningful than gradians or degrees. How many radians make up a circle? 

Watch This

http://www.youtube.com/watch?v=nAJqXtzwpXQ James Sousa: Radian Measure

Guidance

A radian is defined to be the central angle where the subtended arc length is the same length as the radius. 

Another way to think about radians is through the circumference of a circle. The circumference of a circle with radius r  is 2 \pi r . Just over six radii (exactly 2 \pi  radii) would stretch around any circle. 

To define a radian in terms of degrees, equate a circle measured in degrees to a circle measured in radians. 

360 \ degrees=2 \pi \ radians , so \frac{180}{\pi} \ degrees=1 \ radian

Alternatively; 360 \ degrees=2 \pi \ radians , so 1 \ degree= \frac{\pi}{180} \ radians

The conversion factor to convert degrees to radians is: \frac{\pi}{180^\circ}  

The conversion factor to convert radians to degrees is:  \frac{180^\circ}{\pi}

If an angle has no units, it is assumed to be in radians. 

Example A

Convert 150^\circ  into radians. 

Solution:  150^\circ \cdot \frac{\pi}{180^\circ}=\frac{15\pi}{18}=\frac{5 \pi}{6} \ radians

Make sure the degree units cancel. 

Example B

Convert \frac{\pi}{6}  radians into degrees. 

Solution:  \frac{\pi}{6} \cdot \frac{180^\circ}{\pi}=\frac{180^\circ}{6}=30^\circ

Often the \pi ’s will cancel. 

Example C

Convert (6 \pi)^\circ  into radians. 

Solution: Don’t be fooled just because this has \pi . This number is about 19^\circ

(6 \pi)^\circ \cdot \frac{\pi}{180^\circ}=\frac{6 \pi^2}{180}=\frac{\pi^2}{3}

It is very unusual to ever have a \pi^2  term, but it can happen. 

Concept Problem Revisited

Exactly 2 \pi  radians describe a circular arc. This is because 2 \pi  radiuses wrap around the circumference of any circle. 

Vocabulary

A radian is defined to be the central angle where the subtended arc length is the same length as the radius. 

A subtended arc is the part of the circle in between the two rays that make the central angle. 

Guided Practice

1. Convert \frac{5\pi}{6}  into degrees. 

2. Convert 210^\circ  into radians. 

3. Draw a \frac{\pi}{2}  angle by first drawing a 2 \pi  angle, halving it and halving the result. 

Answers:

1.  \frac{5\pi}{6} \cdot \frac{180^\circ}{\pi}=\frac{5 \cdot 30^\circ}{1}=150^\circ

2. 210^\circ \cdot \frac{\pi}{180^\circ}=\frac{7 \cdot 30 \cdot \pi}{6 \cdot 30}=\frac{7\pi}{6}

3.  \frac{\pi}{2}=90^\circ

Practice

Find the radian measure of each angle.

1.  120^\circ

2. 300^\circ

3. 90^\circ

4. 330^\circ

5. 270^\circ

6.  45^\circ

7. (5 \pi)^\circ

Find the degree measure of each angle.

8.  \frac{7 \pi}{6}

9.  \frac{5 \pi}{4}

10.  \frac{3 \pi}{2}

11.  \frac{5 \pi}{3}

12. \pi

13. \frac{\pi}{6}

14. 3

15. Explain why if you are given an angle in degrees and you multiply it by  \frac{\pi}{180} you will get the same angle in radians.

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